Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map,
map_map]
simp only [Function.comp_def, Fin.val_succ]
| Mathlib/Data/List/FinRange.lean | 79 | 82 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... |
variable {N : Type u4} {N' : Type u5} [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N']
/-- The morphism of exterior algebras induced by a linear map. -/
def map (f : M →ₗ[R] N) : ExteriorAlgebra R M →ₐ[R] ExteriorAlgebra R N :=
CliffordAlgebra.map { f with map_app' := fun _ => rfl }
@[simp]
theorem ma... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 351 | 359 |
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Conditional Probability
This file defines conditional probability and includes basic resu... | ⟨by
unfold ProbabilityTheory.cond
| Mathlib/Probability/ConditionalProbability.lean | 150 | 151 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s)
{g : ℝ → ℝ} (hg : ∀ᵐ x, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s))
{f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
(volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
= ENNReal.ofReal ... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,241 | 1,248 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Nonneg.Lattice
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Nonneg/Ring.lean | 362 | 362 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,958 | 1,960 | |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | mv_polynomial σ R →+* mv_polynomial τ S :=
eval₂_hom f g
def join (f : R →+* S) : mv_polynomial (mv_polynomial σ R) S →ₐ[S] mv_polynomial σ S :=
aeval (map f)
def ajoin [algebra R S] : mv_polynomial (mv_polynomial σ R) S →ₐ[S] mv_polynomial σ S :=
join (algebra_map R S)
-/
end MvPolynomial
| Mathlib/Algebra/MvPolynomial/Monad.lean | 352 | 381 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | exact ⟨|q'|, mod_cast (sqrt_mul_self_eq_abs q').symm⟩
theorem irrational_sqrt_ratCast_iff {q : ℚ} :
Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q := by
obtain hq | hq := le_or_lt 0 q
· simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]
· rw [sqrt_eq_zero_of_nonpos (Rat.cast_nonpos.2 hq.le)]
... | Mathlib/Data/Real/Irrational.lean | 103 | 111 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | lemma aeval_comp_rename [Algebra R S] :
(aeval (R := R) g).comp (rename k) = MvPolynomial.aeval (g ∘ k) :=
AlgHom.ext fun p ↦ aeval_rename k g p
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
| Mathlib/Algebra/MvPolynomial/Rename.lean | 199 | 203 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | Mathlib/Analysis/Complex/Basic.lean | 756 | 758 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | /-- If the angle between two vectors is `π`, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : x ≠ 0 :=
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 91 | 92 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` w... | carrier := S
| Mathlib/Analysis/Convex/Cone/Basic.lean | 441 | 441 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | end Image
/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/
instance isSplitMono_sigma_ι {β : Type u'} [HasZeroMorphisms C] (f : β → C)
[HasColimit (Discrete.functor f)] (b : β) : IsSplitMono (Sigma.ι f b) := by
classical exact IsSplitMono.mk' { retraction := Sig... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 546 | 552 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Polynom... | · exact mod_cast Nat.succ_le_iff.mpr (Nat.choose_pos (d.div_le_self 2))
end Roots
end Polynomial
| Mathlib/Topology/Algebra/Polynomial.lean | 198 | 215 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,416 | 1,418 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by
rw [eq_top_iff_one, Submodule.mem_sup]
| Mathlib/RingTheory/Ideal/Operations.lean | 529 | 530 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.PullbackCarrier
import Mathlib.Topology.LocalAtTarget
/-!
# Universally closed ... | constructor
intro X' Y' i₁ i₂ f' H
have := UniversallyClosed.out _ _ _ ((IsPullback.of_hasPullback i₁ f).paste_horiz H)
exact IsClosedMap.of_comp_surjective (MorphismProperty.pullback_fst (P := @Surjective) _ _ ‹_›).1
(Scheme.Hom.continuous _) this
| Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 72 | 76 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.SpecialFuncti... | funext deriv_log
theorem contDiffAt_log {n : WithTop ℕ∞} {x : ℝ} : ContDiffAt ℝ n log x ↔ x ≠ 0 := by
refine ⟨fun h ↦ continuousAt_log_iff.1 h.continuousAt, fun hx ↦ ?_⟩
| Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 67 | 70 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | theorem ascPochhammer_eval₂ (f : S →+* T) (n : ℕ) (t : T) :
(ascPochhammer T n).eval t = (ascPochhammer S n).eval₂ f t := by
rw [← ascPochhammer_map f]
exact eval_map f t
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 83 | 87 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.Calculus.DiffContOnCl
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.F... | obtain ⟨δ, δ0, hδ⟩ : ∃ δ > (0 : ℝ), ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π) :=
((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _
(div_pos ε0 Real.two_pi_pos)
obtain ⟨r, hr0, hrδ, hrR⟩ : ∃ r, 0 < r ∧ r ≤ δ ∧ r ≤ R :=
⟨min δ R, lt_min δ0 h0, min_le_left _ ... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 350 | 392 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | lemma holderConjugate_iff_eq_conjExponent (hp : 1 < p) : p.HolderConjugate q ↔ q = p / (p - 1) :=
⟨HolderConjugate.conjugate_eq, fun h ↦ holderConjugate_iff.mpr ⟨hp, by field_simp [h]⟩⟩
| Mathlib/Data/Real/ConjExponents.lean | 194 | 195 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.Basic
import Mathlib.RingTheory.Ideal.Quotient.Operations
/-!
# Results
- `derivationToSquareZeroOfLift`: The ... |
@[simp]
theorem liftOfDerivationToSquareZero_mk_apply' (d : Derivation R A I) (x : A) :
(Ideal.Quotient.mk I) (d x) + (algebraMap A (B ⧸ I)) x = algebraMap A (B ⧸ I) x := by
simp only [Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add]
| Mathlib/RingTheory/Derivation/ToSquareZero.lean | 106 | 110 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | noncomputable def P'IsoUnopOpP : S.P' ≅ Opposite.unop S.op.P := pushoutIsoUnopPullback _ _
lemma op_δ : S.op.δ = S.δ.op := Quiver.Hom.unop_inj (by
rw [Quiver.Hom.unop_op, ← cancel_mono (pushout.inr _ _ : _ ⟶ S.P'),
← cancel_epi (pullback.snd _ _ : S.P ⟶ _), S.snd_δ_inr,
← cancel_mono S.P'IsoUnopOpP.hom, ← ca... | Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 336 | 344 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Kim Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.Products.Basic
/-!
# Categories of indexed families... | -/
@[simps]
def pi' {E : Type*} [Category E] {F G : E ⥤ ∀ i, C i}
(τ : ∀ i, F ⋙ Pi.eval C i ⟶ G ⋙ Pi.eval C i) : F ⟶ G where
app := fun X i => (τ i).app X
naturality _ _ f := by
ext i
exact (τ i).naturality f
end NatTrans
namespace NatIso
variable {C}
| Mathlib/CategoryTheory/Pi/Basic.lean | 246 | 259 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | ⟨IsRelPrime.isCoprime, IsCoprime.isRelPrime⟩
variable (R)
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 206 | 209 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalBasis b) {a : α} :
(𝓝 a).HasBasis (fun t : Set α => t ∈ b ∧ a ∈ t) fun t => t :=
⟨fun s => hb.mem_nhds_iff.trans <| by simp only [and_assoc]⟩
protected theorem IsTopologicalBasis.isOpen {s : Set α} {b : Set (Set α)}
(hb : IsTopolo... | Mathlib/Topology/Bases.lean | 143 | 153 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... |
lemma rpow_sub_natCast' (hx : 0 ≤ x) (h : y - n ≠ 0) : x ^ (y - n) = x ^ y / x ^ n := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 427 | 428 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
/-!
# Variance of random ... | lemma evariance_ne_top [IsFiniteMeasure μ] (hX : MemLp X 2 μ) : evariance X μ ≠ ∞ :=
(evariance_lt_top hX).ne
theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ) (hX : ¬MemLp X 2 μ) :
evariance X μ = ∞ := by
by_contra h
| Mathlib/Probability/Variance.lean | 89 | 94 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,357 | 1,358 | |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# Squares and even elements
This ... |
section DivisionMonoid
variable [DivisionMonoid α] {a : α}
| Mathlib/Algebra/Group/Even.lean | 141 | 143 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,233 | 2,239 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,300 | 1,302 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 1,254 | 1,256 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
/-!
# Schwarz lemma
In this file we prove several versions of the Schw... | ‖dslope f c z‖ ≤ R₂ / R₁ := by
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
suffices ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r by
refine ge_of_tendsto ?_ this
exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds
rw [mem_ball] at hz
filter_upwards [Ioo_mem_nhdsLT hz] with r ... | Mathlib/Analysis/Complex/Schwarz.lean | 65 | 88 |
/-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer
-/
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Combinatorics.SimpleG... | p.IsHamiltonianCycle ↔ p.IsCycle ∧ p.tail.IsHamiltonian :=
⟨fun ⟨h, h'⟩ ↦ ⟨h, h'⟩, fun ⟨h, h'⟩ ↦ ⟨h, h'⟩⟩
lemma isHamiltonianCycle_iff_isCycle_and_support_count_tail_eq_one :
| Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | 91 | 94 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | theorem sup_multiset_prod_eq_top {s : Multiset (Ideal R)} (h : ∀ p ∈ s, I ⊔ p = ⊤) :
I ⊔ Multiset.prod s = ⊤ :=
Multiset.prod_induction (I ⊔ · = ⊤) s (fun _ _ hp hq ↦ (sup_mul_eq_of_coprime_left hp).trans hq)
| Mathlib/RingTheory/Ideal/Operations.lean | 586 | 588 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.... |
/-- If every dissociated subset of `s` has size at most `d`, then `s` is actually generated by a
subset of size at most `d`.
This is a dissociation analog of the fact that a set whose linearly independent subsets all have
size at most `d` is of dimension at most `d` itself. -/
@[to_additive "If every dissociated subs... | Mathlib/Combinatorics/Additive/Dissociation.lean | 119 | 155 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | Mathlib/Logic/Function/Basic.lean | 1,067 | 1,069 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | exact hs hx hy (hfinj.ne hxy) ha hb hab
theorem StrictConvex.is_linear_preimage {s : Set F} (hs : StrictConvex 𝕜 s) {f : E → F}
(h : IsLinearMap 𝕜 f) (hf : Continuous f) (hfinj : Injective f) :
StrictConvex 𝕜 (s.preimage f) :=
hs.linear_preimage (h.mk' f) hf hfinj
| Mathlib/Analysis/Convex/Strict.lean | 130 | 135 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | simp [ha.multiplicity_eq_iff, ha.not_dvd_of_one_right]
@[deprecated (since := "2024-11-30")]
alias multiplicity.Finite.one_right := FiniteMultiplicity.one_right
theorem FiniteMultiplicity.not_of_unit_left (a : α) (u : αˣ) : ¬ FiniteMultiplicity (u : α) a :=
FiniteMultiplicity.not_of_isUnit_left a u.isUnit
@[depr... | Mathlib/RingTheory/Multiplicity.lean | 381 | 399 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | rw [← cancel_mono i₂, isoOfMkEqMk_hom, ofMkLEMk_comp, fac]
lemma mk_lt_mk_iff_of_comm {X A₁ A₂ : C} {i₁ : A₁ ⟶ X} {i₂ : A₂ ⟶ X} [Mono i₁] [Mono i₂]
| Mathlib/CategoryTheory/Subobject/Basic.lean | 449 | 451 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | theorem bsupr_limsup_dimH (s : Set X) : ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets = dimH s := by
refine le_antisymm (iSup₂_le fun x _ => ?_) ?_
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 231 | 232 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 1,516 | 1,517 | |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... |
theorem hom.faithful : ∀ c d, Function.Injective fun f : c ⟶ d => (hom S).map f := by
rintro ⟨c, hc⟩ ⟨d, hd⟩ ⟨f, hf⟩ ⟨g, hg⟩ hfg; exact Subtype.eq hfg
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 152 | 154 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 253 | 253 | |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod]
/-- Any `n`-th primitive root of unity is a root of `cyclotomic n R`. -/
theorem _root_.IsPrimitiveRoot.isRoot_cyclotomic (hpos : 0 < n) {μ : R} (h : IsPrimitiveRoot μ n) :
| Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 56 | 59 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | theorem hasFDerivAt_fourierIntegral
(hf_int : Integrable f) (hvf_int : Integrable (fun v ↦ ‖v‖ * ‖f v‖)) (x : V) :
HasFDerivAt (𝓕 f) (𝓕 (fourierSMulRight (innerSL ℝ) f) x) x :=
VectorFourier.hasFDerivAt_fourierIntegral (innerSL ℝ) hf_int hvf_int x
| Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 674 | 678 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | apply M'.mem_of_span_top_of_smul_mem _ (Ideal.span_range_pow_eq_top s hs f)
rintro ⟨_, r, hr, rfl⟩
exact hf r
open Pointwise in
@[simp]
theorem map_pointwise_smul (r : R) (N : Submodule R M) (f : M →ₗ[R] M') :
(r • N).map f = r • N.map f := by
simp_rw [← ideal_span_singleton_smul, map_smul'']
| Mathlib/RingTheory/Ideal/Operations.lean | 162 | 171 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | rw [Fin.natCast_eq_last]
exact Fin.le_last i
| Mathlib/Data/Fin/Basic.lean | 389 | 390 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | dsimp
exact h_mul _ _ _ _ _ _
| Mathlib/RingTheory/TensorProduct/Basic.lean | 703 | 705 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | calc
_ = ∫ x, (pdf X ℙ volume x).toReal * x := by congr with x; exact mul_comm _ _
_ = _ := integral_pdf_smul measurable_id.aestronglyMeasurable
theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
(hgi : ∫⁻ x, ‖f x‖ₑ * g x ≠ ∞) :
HasFiniteIntegral fun x => f x * (pdf ... | Mathlib/Probability/Density.lean | 279 | 294 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | ofReal_le_natCast
| Mathlib/Data/ENNReal/Real.lean | 219 | 219 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 793 | 798 | |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | @[simp]
theorem basis_apply (h : IsAdjoinRootMonic S f) (i) : h.basis i = h.root ^ (i : ℕ) :=
| Mathlib/RingTheory/IsAdjoinRoot.lean | 403 | 404 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 733 | 734 | |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 555 | 560 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | (hg : ∀ x, x ∈ s ↔ g x ∈ s) (hc : c.support ⊆ s) (n : ℤ) :
c ^ n = ofSubtype (g.subtypePerm hg) ↔
c.subtypePerm (isInvariant_of_support_le hc) ^ n = g.subtypePerm hg := by
constructor
· intro h
ext ⟨x, hx⟩
simp only [Perm.congr_fun h x, subtypePerm_apply_zpow_of_mem, Subtype.coe_mk, subtypePer... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 1,058 | 1,072 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
/-!
# Simplicial complexes
In this file, we define ... | /-- The complex consisting of only the faces present in both of its arguments. -/
instance : Min (SimplicialComplex 𝕜 E) :=
⟨fun K L =>
{ faces := K.faces ∩ L.faces
not_empty_mem := fun h => K.not_empty_mem (Set.inter_subset_left h)
indep := fun hs => K.indep hs.1
down_closed := fun hs hst ht =... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 204 | 212 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,479 | 1,483 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | eLpNorm' f q (0 : Measure α) = ∞ := by simp [eLpNorm', hq_neg]
end ENormedAddMonoid
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 146 | 149 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | Measurable (fun p : α × γ ↦ Kernel.rnDerivAux κ η p.1 p.2) := by
simp_rw [rnDerivAux]
split_ifs with hα
· refine Measurable.ennreal_toReal ?_
change Measurable ((fun q : γ × α ↦ (κ q.2).rnDeriv (η q.2) q.1) ∘ Prod.swap)
refine (measurable_from_prod_countable' (fun a ↦ ?_) ?_).comp measurable_swap
... | Mathlib/Probability/Kernel/RadonNikodym.lean | 112 | 127 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | rw [Nat.add_comm]; symm; apply dropn_add
@[simp]
theorem head_dropn (s : Seq α) (n) : head (drop s n) = get? s n := by
induction' n with n IH generalizing s; · rfl
rw [← get?_tail, ← dropn_tail]; apply IH
@[simp]
theorem drop_succ_cons {x : α} {s : Seq α} {n : ℕ} :
(cons x s).drop (n + 1) = s.drop n := by
... | Mathlib/Data/Seq/Seq.lean | 1,030 | 1,045 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | /-! ### Injectivity and surjectivity lemmas for image and preimage -/
section ImagePreimage
| Mathlib/Data/Set/Image.lean | 1,282 | 1,285 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | (gaussianPDF μ v x).toReal = gaussianPDFReal μ v x := by
rw [gaussianPDF, ENNReal.toReal_ofReal (gaussianPDFReal_nonneg μ v x)]
| Mathlib/Probability/Distributions/Gaussian.lean | 169 | 171 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | · exact cycleRange_of_lt h₁
· exact cycleRange_of_eq h₂
· exact cycleRange_of_gt (lt_of_le_of_ne (le_of_not_gt h₁) (Ne.symm h₂))
@[simp]
theorem cycleRange_zero (n : ℕ) [NeZero n] : cycleRange (0 : Fin n) = 1 := by
ext j
rcases (Fin.zero_le' j).eq_or_lt with rfl | hj
· simp
· rw [cycleRange_of_gt hj, one... | Mathlib/GroupTheory/Perm/Fin.lean | 193 | 202 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 262 | 265 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,451 | 1,457 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | @[to_additive (attr := simp)]
lemma map_comp_mul [MulHomClass F M N] (f : F) (g h : ι → M) : f ∘ (g * h) = f ∘ g * f ∘ h := by
ext; simp
| Mathlib/Algebra/Group/Hom/Defs.lean | 314 | 316 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | · rintro ⟨⟨⟨k, rfl⟩, hn⟩, rfl⟩
rw [Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt]
exact ⟨rfl, hn⟩
· rintro ⟨rfl, hn⟩
exact ⟨⟨dvd_mul_right _ _, hn⟩, Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt⟩
| Mathlib/NumberTheory/Divisors.lean | 359 | 364 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,196 | 1,198 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | refine (cancel_epi ((tensorLeft _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 ?_
dsimp [X]
slice_lhs 1 2 => rw [← comp_whiskerRight, whiskerLeft_π_actLeft, comp_whiskerRight,
comp_whiskerRight]
slice_lhs 3 4 => rw [π_tensor_id_actRight]
slice_lhs 2 3 => rw [associator_naturality_left]
-- Porting note: ... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 317 | 325 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
/-! # Euler's infinite product for the sine function
This file... | convert (tendsto_mul_iff_of_ne_zero _ one_ne_zero).mp A
suffices Tendsto (fun n : ℕ =>
(∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) /
(∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ)) atTop (𝓝 1) from
this.comp (tendsto_id.const_mul_atTop' zero_lt_two)
have : ContinuousOn (fun x... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 299 | 321 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... |
/-- The topology of uniform convergence indeed gives the same notion of convergence as
`TendstoUniformly`. -/
protected theorem tendsto_iff_tendstoUniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (toFun ∘ F) (toFun f) p := by
rw [(UniformFun.hasBasis_nhds α β f).tendsto_right_iff, ... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 448 | 453 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | · simp only [Matrix.mul_apply, Matrix.conjTranspose_apply, star_def, PiLp.inner_apply,
inner_apply']
congr
· rw [orthonormal_iff_ite.mp b.orthonormal i j]
rfl
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 746 | 751 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,567 | 1,570 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... |
@[simp]
| Mathlib/Data/Real/Sqrt.lean | 308 | 309 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | theorem digits_add (b : ℕ) (h : 1 < b) (x y : ℕ) (hxb : x < b) (hxy : x ≠ 0 ∨ y ≠ 0) :
digits b (x + b * y) = x :: digits b y := by
rcases Nat.exists_eq_add_of_le' h with ⟨b, rfl : _ = _ + 2⟩
cases y
| Mathlib/Data/Nat/Digits.lean | 124 | 127 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 439 | 444 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | point in a domain. -/
theorem ContinuousLinearEquiv.comp_contDiffWithinAt_iff (e : F ≃L[𝕜] G) :
ContDiffWithinAt 𝕜 n (e ∘ f) s x ↔ ContDiffWithinAt 𝕜 n f s x :=
⟨fun H => by
simpa only [Function.comp_def, e.symm.coe_coe, e.symm_apply_apply] using
H.continuousLinearMap_comp (e.symm : G →L[𝕜] F),
... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 293 | 301 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# Matrix results for barycentric co-ordinates
Results about the ... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 170 | 174 | |
/-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer
-/
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Combinatorics.SimpleG... | /-- A path whose support contains every vertex is hamiltonian. -/
lemma IsPath.isHamiltonian_of_mem (hp : p.IsPath) (hp' : ∀ w, w ∈ p.support) :
p.IsHamiltonian := fun _ ↦
le_antisymm (List.nodup_iff_count_le_one.1 hp.support_nodup _) (List.count_pos_iff.2 (hp' _))
| Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | 46 | 50 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
/-!
# Lemmas which are consequences of monoidal coher... |
@[reassoc]
theorem leftUnitor_tensor' (X Y : C) :
| Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 30 | 32 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,039 | 1,045 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | hr.radius_pos
theorem HasFPowerSeriesWithinOnBall.of_le
(hf : HasFPowerSeriesWithinOnBall f p s x r) (r'_pos : 0 < r') (hr : r' ≤ r) :
| Mathlib/Analysis/Analytic/Basic.lean | 517 | 520 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | exact InfinitePlace.mult_comap_le _ _
variable (K)
variable [Algebra K F] [IsScalarTower k K F]
lemma IsUnramified.of_restrictScalars {w : InfinitePlace F} (h : IsUnramified k w) :
IsUnramified K w := by
rw [InfinitePlace.isUnramified_iff_mult_le, ← h.eq, IsScalarTower.algebraMap_eq k K F,
InfinitePlace.c... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 821 | 835 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 490 | 501 | |
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + ... | · intro k
rw [hyperoperation_three]
dsimp
rw [one_pow]
· intro k
| Mathlib/Data/Nat/Hyperoperation.lean | 91 | 95 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | theorem natCast_eq_zero_of_even_of_two_eq_zero {n : ℕ} (hn : Even n) (h : (2 : R) = 0) :
(n : R) = 0 :=
| Mathlib/Algebra/Ring/Parity.lean | 375 | 376 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... | Note that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/
instance fintypeRange [Fintype A] [DecidableEq B] (φ : A →ₐ[R] B) : Fintype φ.range :=
Set.fintypeRange φ
end AlgHom
| Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 563 | 567 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | div_eq_mul_inv s t ▸ hs.mul ht.inv
end SeminormedGroup
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 47 | 49 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | rw [uniqueBaseOn, freeOn_restrict rfl.subset]
@[simp] theorem uniqueBaseOn_empty (I : Set α) : uniqueBaseOn ∅ I = loopyOn I := by
| Mathlib/Data/Matroid/Constructions.lean | 242 | 244 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | instance (priority := 10) GroupWithZero.noZeroDivisors : NoZeroDivisors G₀ :=
{ (‹_› : GroupWithZero G₀) with
eq_zero_or_eq_zero_of_mul_eq_zero := @fun a b h => by
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 229 | 231 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | apply sum_congr rfl
intro x hx
simp only [log_injOn_pos (hz' j hj) (hz' x hx), h j hj, h x hx]
/-- **AM-GM inequality - equality condition**: This theorem provides the equality condition for the
weighted version of the AM-GM inequality for real-valued nonnegative functions. -/
theorem geom_mean... | Mathlib/Analysis/MeanInequalities.lean | 230 | 236 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Category.Cat.AsSmall
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.IsConnected
i... | [PreservesColimit (F ⋙ G) H] : PreservesColimit G H where
preserves {c} hc := by
refine ⟨isColimitWhiskerEquiv F _ ?_⟩
let hc' := isColimitOfPreserves H ((isColimitWhiskerEquiv F _).symm hc)
exact IsColimit.ofIsoColimit hc' (Cocones.ext (Iso.refl _) (by simp))
theorem reflectsColimit_of_comp {B : Typ... | Mathlib/CategoryTheory/Limits/Final.lean | 386 | 404 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Semirings and ... |
section
| Mathlib/Algebra/Ring/Semiconj.lean | 39 | 41 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | ⟨(x, y), ⟨hx, hy⟩⟩
theorem Nonempty.fst : (s ×ˢ t).Nonempty → s.Nonempty := fun ⟨x, hx⟩ => ⟨x.1, hx.1⟩
| Mathlib/Data/Set/Prod.lean | 261 | 263 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | tfae_have 3 → 1 := by
refine fun h => Acc.intro _ (fun b hb => (h.apply ⟨b, .single hb⟩).of_fibration Subtype.val ?_)
exact fun ⟨c, hc⟩ d h => ⟨⟨d, .head h hc⟩, h, rfl⟩
tfae_finish
end WellFoundedOn
end AnyRel
section IsStrictOrder
variable [IsStrictOrder α r] {s t : Set α}
instance IsStrictOrder.subse... | Mathlib/Order/WellFoundedSet.lean | 147 | 162 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,385 | 3,387 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | rw [Nat.Ico_succ_right, Nat.Icc_succ_left, Nat.cast_succ]
_ ≤ ((k + 1 : α) ^ 2)⁻¹ + (k + 1 : α)⁻¹ := by
refine add_le_add le_rfl ((sum_Ioc_inv_sq_le_sub ?_ (le_max_left _ _)).trans ?_)
· simp only [Ne, Nat.succ_ne_zero, not_false_iff]
· simp only [Nat.cast_succ, one_div, sub_le_self_iff, inv... | Mathlib/Analysis/PSeries.lean | 401 | 418 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... |
theorem pow_dvd_of_le_multiplicity {k : ℕ} (hk : k ≤ multiplicity a b) :
a ^ k ∣ b := pow_dvd_of_le_emultiplicity (le_emultiplicity_of_le_multiplicity hk)
| Mathlib/RingTheory/Multiplicity.lean | 258 | 260 |
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